Lab #$1$

Using the Power Rule to find equations to a tangent line

Find the equation of the line tangent to the graphs of $f(x)=\frac{-5}{\sqrt[\phantom{2}]{x}}$ and $f(x)=\frac{-5+\sqrt[\phantom{2}]{x}}{\sqrt[\phantom{2}]{x}}$ at $x=4.$ Graph both functions and tangent lines when $x=4.$ What can you say about the two lines? For what values of $x$ does the tangent line have slope of $0,$ if any?

Graph $f,f^{\text{'}},f^{\text{'}\text{'}}$ on the same set of axes for $f(x)=x^3+x^2-5x+1.$

a$.$ For what values of $x$ does the tangent line of $f$ have a slope of $0,$ if any?

b$.$ For what values of $x$ is $f^{\text{'}\text{'}}=0?$ What is happening to $f$ at that same value?

c$.$ For what intervals is $f$ increasing? For what intervals is $f^{\text{'}}$ positive?

Graph $f(x)=x^2$ and $g(x)=\sqrt[\phantom{2}]{x}$ on the same set of axes. Find and graph the equation of the tangent line of $f$ when $x=3$ and of $g$ when $x=9.$ What do you notice about the equations of the tangent lines?